Affiliation:
1. Department of Mathematics and Data Science, Faculty of Mathematics and Natural Science, Andalas University, Kampus Unand Limau Manis, Padang 25163, Indonesia
Abstract
A picture in a group presentation is a geometric configuration with an arrangement of discs and arcs within a boundary disc. The drawing of this picture does not have to follow a particular rule, only using the generator as discs and the relation as arcs. It will form a picture label pattern if drawn with a particular rule. This paper discusses the label pattern of a picture in the presentation of direct product groups. Direct product presentation is used with two cyclic groups,
and
where
and
. The method for forming a picture label pattern is to arrange the first generator in the initial arrangement, compile a second generator, and add a number of commutators. Furthermore, the pattern is used to calculate the length of the label on the picture. It is obtained that the picture’s label is
and the length of the label is
, where
is the number of commutator discs.